Cluster Structures on Double Bott-Samelson Cells

Let \(C\) be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to \(C\). We prove that the decorated double Bott-Samelson cells are smooth affine varieties, whose c...

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Bibliographic Details
Published in:arXiv.org 2022-04
Main Authors: Shen, Linhui, Weng, Daping
Format: Article
Language:English
Subjects:
Braid theory
Clusters
Mathematical analysis
Matrix methods
Periodic variations
Rings (mathematics)
Transformations
Online Access:Get full text
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Summary:Let \(C\) be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to \(C\). We prove that the decorated double Bott-Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras. We explicitly describe the Donaldson-Thomas transformations on double Bott-Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock-Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson-Thomas transformations on a family of double Bott-Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov's periodicity conjecture in the cases of \(\Delta\square \mathrm{A}_r\). When \(C\) is of type \(\mathrm{A}\), the double Bott-Samelson cells are isomorphic to Shende-Treumann-Zaslow's moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their \(\mathbb{F}_q\)-points we obtain rational functions which are Legendrian link invariants.
ISSN:2331-8422